4. Let f1, f2, fz be functions in F(R). a. For a set of real numbers x1, X2, X3, let (F(x;)) be the 3-by-3 matrix whose (i, j) entry is f(x;), for 1< i, j s 3. Prove that the functions f1, f2, f3 are linearly independent if the rows of the matrix (f(x})) are linearly independent. b. Assume the functions f1, f2, f3 have first and second derivatives on some interval (a, b), and let W(x) be the 3-by-3 matrix whose (i, j) entry is fY-", for 1 < i, j< 3, where f'0) = f, f(1) = f' and f(2) = f" for a dif- ferentiable function f. Prove that f1, f2, f3 are linearly independent if for some x in (a, b), the rows of the matrix W(x) are linearly independent. Show that the following sets of functions are linearly independent. %3D c. f;(x) = –x² + x + 1, f2(x) = x² + 2x, f3(x) = x² – 1. d. fi(x) = e¯*, f2(x) = x, f3(x) = e²ª. e. f,(x) = e*, f2(x) = sin x, f3(x) Note that if the tests in (a) or (b) fail, it is not guaranteed that the functions are linearly dependent. %3D = cos x.
Percentage
A percentage is a number indicated as a fraction of 100. It is a dimensionless number often expressed using the symbol %.
Algebraic Expressions
In mathematics, an algebraic expression consists of constant(s), variable(s), and mathematical operators. It is made up of terms.
Numbers
Numbers are some measures used for counting. They can be compared one with another to know its position in the number line and determine which one is greater or lesser than the other.
Subtraction
Before we begin to understand the subtraction of algebraic expressions, we need to list out a few things that form the basis of algebra.
Addition
Before we begin to understand the addition of algebraic expressions, we need to list out a few things that form the basis of algebra.
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